Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i e_i$, For which pairs $(B, \{e_i\}_{i=1}^\infty)$, $p_n$ convergence is equivalent to $||p_n||$ convergence? It is obviously truly for $\ell_1$ and $\ell_2$ with standart basises, but maybe it is exist abstract characterization?
I created another question before Convergence of partial sums of basis vectors in banach space. But this question is slightly wider then question on link, and I dont think it is ethical to put off "right answer" flag from beautiful David Ulrich example and edit my previous question.
Upd: My assumption that sufficient condition - it is equivalence of basis $\{e_i\}_{i=1}^\infty$ to standart basis in $\ell_1$ it is true?